Very thin, very expensive, very low atomic mass number beryllium windows.

I'm not an expert but I believe that the particles are traveling so fast and with so much energy that they just blast through the walls of the vacuum chamber

There's probably a material which photons and other sub atomic particles can pass through but the molecules in the air can't.

For some applications a plasma window can be used.

Probably just a very small hole and beefing up the vacuum equipment near it. From what I recall the LHC beam is on the scale of a couple micrometres.

No you just fire it through the wall of the vacuum chamber. When you're going >99% the speed of light a couple inches of metal don't matter that much anymore.

Why? Just do it all under vacuum.

But in cases where you must (cost/practicality limitations, dealing with liquids, etc), this is done by passing through a thin film of material that has a small cross section of interaction with the particles, typically "low Z" (low atomic number) elements like Beryllium (its rigidity makes withstanding 1 atm pressure difference easy), or simple hydrocarbon plastics like polyethylene/polypropylene.

But this depends on what particles you're talking about.